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Publications (10 of 21) Show all publications
Yurova, E. & Khrennikov, A. (2019). Description of (Fully) Homomorphic Cryptographic Primitives Within the p-Adic Model of Encryption. In: Karl‐Olof Lindahl, Torsten Lindström, Luigi G. Rodino, Joachim Toft, Patrik Wahlberg (Ed.), Analysis, Probability, Applications, and Computation: Proceedings of the 11th ISAAC Congress, Växjö (Sweden) 2017. Paper presented at 11th ISAAC Congress, Växjö, Sweden, 2017 (pp. 241-248). Cham: Birkhäuser Verlag
Open this publication in new window or tab >>Description of (Fully) Homomorphic Cryptographic Primitives Within the p-Adic Model of Encryption
2019 (English)In: Analysis, Probability, Applications, and Computation: Proceedings of the 11th ISAAC Congress, Växjö (Sweden) 2017 / [ed] Karl‐Olof Lindahl, Torsten Lindström, Luigi G. Rodino, Joachim Toft, Patrik Wahlberg, Cham: Birkhäuser Verlag, 2019, p. 241-248Conference paper, Published paper (Refereed)
Abstract [en]

In this paper we consider a description of homomorphic and fully homomorphic cryptographic primitives in the p-adic model. This model describes a wide class of ciphers (including substitution ciphers, substitution ciphers streaming, keystream ciphers in the alphabet of p elements), but certainly not all. Homomorphic and fully homomorphic ciphers are used to ensure the credibility of remote computing, including cloud technology. Within considered p-adic model we describe all homomorphic cryptographic primitives with respect to arithmetic and coordinate-wise logical operations in the ring of p-adic integers ℤ p . We show that there are no fully homomorphic cryptographic primitives for each pair of the considered set of arithmetic and coordinate-wise logical operations on ℤ p.

Place, publisher, year, edition, pages
Cham: Birkhäuser Verlag, 2019
Series
Trends in Mathematics, ISSN 2297-0215, E-ISSN 2297-024X
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-82771 (URN)10.1007/978-3-030-04459-6_23 (DOI)2-s2.0-85065438988 (Scopus ID)978-3-030-04458-9 (ISBN)978-3-030-04459-6 (ISBN)
Conference
11th ISAAC Congress, Växjö, Sweden, 2017
Available from: 2019-05-27 Created: 2019-05-27 Last updated: 2019-08-29Bibliographically approved
Yurova, E. (2019). On the Injective Embedding of p-Adic Integers in the Cartesian Product of p Copies of Sets of 2-Adic Integers. In: Karl‐Olof Lindahl, Torsten Lindström, Luigi G. Rodino, Joachim Toft, Patrik Wahlberg (Ed.), Analysis, Probability, Applications, and Computation: . Paper presented at 11th ISAAC Congress, Växjö (Sweden, 2017 (pp. 233-239). Birkhäuser Verlag
Open this publication in new window or tab >>On the Injective Embedding of p-Adic Integers in the Cartesian Product of p Copies of Sets of 2-Adic Integers
2019 (English)In: Analysis, Probability, Applications, and Computation / [ed] Karl‐Olof Lindahl, Torsten Lindström, Luigi G. Rodino, Joachim Toft, Patrik Wahlberg, Birkhäuser Verlag, 2019, p. 233-239Conference paper, Published paper (Refereed)
Abstract [en]

We study an injective embedding of p-adic integers in the Cartesian product of p copies of sets of 2-adic integers. This embedding allows to explicitly specify any p-adic integer through p specially selected 2-adic numbers. This representation can be used in p-adic mathematical physics, for example, in justifying choice of the parameter p.

Place, publisher, year, edition, pages
Birkhäuser Verlag, 2019
Series
Trends in Mathematics, ISSN 2297-0215, E-ISSN 2297-024X
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-82715 (URN)10.1007/978-3-030-04459-6_22 (DOI)2-s2.0-85065443830 (Scopus ID)978-3-030-04458-9 (ISBN)978-3-030-04459-6 (ISBN)
Conference
11th ISAAC Congress, Växjö (Sweden, 2017
Available from: 2019-05-27 Created: 2019-05-27 Last updated: 2019-08-29Bibliographically approved
Yurova, E. (2018). On the sub-coordinate representation of p-adic functions. In: Alain Escassut, Cristina Perez-Garcia, Khodr Shamseddine (Ed.), Advances in Ultrametric Analysis: . Paper presented at 14th International Conference on p-adic Functional Analysis, held from June 30–July 4, 2016, at the Université d'Auvergne, Aurillac, France (pp. 285-290). USA: American Mathematical Society (AMS), 704
Open this publication in new window or tab >>On the sub-coordinate representation of p-adic functions
2018 (English)In: Advances in Ultrametric Analysis / [ed] Alain Escassut, Cristina Perez-Garcia, Khodr Shamseddine, USA: American Mathematical Society (AMS), 2018, Vol. 704, p. 285-290Conference paper, Published paper (Refereed)
Abstract [en]

In this paper we introduce a new way of representation of p-adic functions, namely, the sub-coordinate representation. The main feature of such representation is that the values of a function f are given in the canonical form of representation of p-adic number. In the sub-coordinate representation thefunction f is determined by a set of p-valued functions that map a set {0,1,...,p - 1} into itself, and by the order of these functions. As one of the applications ofthe sub-coordinate representation, we study a problem of generalization of Hensel's lifting lemma.

Place, publisher, year, edition, pages
USA: American Mathematical Society (AMS), 2018
Series
Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627
National Category
Other Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-72073 (URN)10.1090/conm/704/14173 (DOI)000434495300016 ()2-s2.0-85047664054 (Scopus ID)978-1-4704-3491-5 (ISBN)978-1-4704-4676-5 (ISBN)
Conference
14th International Conference on p-adic Functional Analysis, held from June 30–July 4, 2016, at the Université d'Auvergne, Aurillac, France
Available from: 2018-04-03 Created: 2018-04-03 Last updated: 2019-05-24Bibliographically approved
Yurova, E. & Khrennikov, A. (2018). Subcoordinate Representation of p-adic Functions and Generalization of Hensel's Lemma. Izvestiya. Mathematics, 82(3), 632-645
Open this publication in new window or tab >>Subcoordinate Representation of p-adic Functions and Generalization of Hensel's Lemma
2018 (English)In: Izvestiya. Mathematics, ISSN 1064-5632, E-ISSN 1468-4810, Vol. 82, no 3, p. 632-645Article in journal (Refereed) Published
Abstract [en]

In this paper we describe a new representation of p-adic functions, the so-called subcoordinate representation. The main feature of the subcoordinaterepresentation of a p-adic function is that the values of the function f are given in the canonical form of the representation of p-adic numbers. The function f itself is determined by a tuple of p-valued functions from the set {0, 1,..., p-1} into itself and by the order in which these functions are used to determine the values of f. We also give formulae that enable one to pass from the subcoordinate representation of a 1-Lipschitz function to its van der Put series representation. The effective use of the subcoordinate representation of p-adic functions is illustrated by a study of the feasibility of generalizing Hensel's lemma.

Place, publisher, year, edition, pages
Russian Academy of Sciences, 2018
Keywords
p-adic numbers; Lipschitz functions; coordinate representation; van der Put series
National Category
Other Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-61501 (URN)10.1070/IM8578 (DOI)000437922000010 ()2-s2.0-85049841451 (Scopus ID)
Available from: 2017-03-21 Created: 2017-03-21 Last updated: 2019-08-29Bibliographically approved
Khrennikov, A. & Yurova, E. (2017). Automaton model of protein: Dynamics of conformational and functional states. Progress in Biophysics and Molecular Biology, 130(A), 2-14
Open this publication in new window or tab >>Automaton model of protein: Dynamics of conformational and functional states
2017 (English)In: Progress in Biophysics and Molecular Biology, ISSN 0079-6107, E-ISSN 1873-1732, Vol. 130, no A, p. 2-14Article in journal (Refereed) Published
Abstract [en]

In this conceptual paper we propose to explore the analogy between ontic/epistemic description of quantum phenomena and interrelation between dynamics of conformational and functional states of proteins. Another new idea is to apply theory of automata to model the latter dynamics. In our model protein's behavior is modeled with the aid of two dynamical systems, ontic and epistemic, which describe evolution of conformational and functional states of proteins, respectively. The epistemic automaton is constructed from the ontic automaton on the basis of functional (observational) equivalence relation on the space of ontic states. This reminds a few approaches to emergent quantum mechanics in which a quantum (epistemic) state is treated as representing a class of prequantum (ontic) states. This approach does not match to the standard protein structure-function paradigm. However, it is perfect for modeling of behavior of intrinsically disordered proteins. Mathematically space of protein's ontic states (conformational states) is modeled with the aid of p-adic numbers or more general ultrametric spaces encoding the internal hierarchical structure of proteins. Connection with theory of p-adic dynamical systems is briefly discussed.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Automaton-model; Conformational and functional states; Proteins; Quantum-like model; Structure-function paradigm
National Category
Biophysics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-61120 (URN)10.1016/j.pbiomolbio.2017.02.003 (DOI)000423002900002 ()28214530 (PubMedID)2-s2.0-85013460624 (Scopus ID)
Projects
Modeling of Complex Hierarchic systemsEU-project Quantum Information Access and Retrieval Theory (QUARTZ), Grant No. 721321
Funder
EU, European Research Council, 721321
Available from: 2017-03-07 Created: 2017-03-07 Last updated: 2019-08-29Bibliographically approved
Yurova Axelsson, E. & Khrennikov, A. (2016). Generalization of Hensel's lemma: Finding the roots of p-adic Lipschitz functions. Journal of Number Theory, 158, 217-233
Open this publication in new window or tab >>Generalization of Hensel's lemma: Finding the roots of p-adic Lipschitz functions
2016 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 158, p. 217-233Article in journal (Refereed) Published
Abstract [en]

In this paper we consider the problem of finding the roots of p-adic functions. In the case, where the function is defined by a polynomial with integer p-adic coefficients, using Hensel's lifting lemma helps us find the roots of the p-adic function.

We generalize Hensel's lifting lemma for a wider class of p  -adic functions, namely, the functions which satisfy the Lipschitz condition with constant , in particular, the functions of this class may be non-differentiable. The paper also presents an iterative procedure for finding approximate (in p  -adic metric) values of the root of pα-Lipschitz functions, thus generalizing the p-adic analogue of Newton's method for such a class of functions.

Keywords
p-Adics; Hensel's lifting lemma; Lipschitz function; Van der Put series
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-46723 (URN)10.1016/j.jnt.2015.06.004 (DOI)000362625500012 ()2-s2.0-84939426535 (Scopus ID)
Available from: 2015-10-12 Created: 2015-10-12 Last updated: 2019-02-22Bibliographically approved
Yurova Axelsson, E. (2015). On the representation of the genetic code by the attractors of 2-adic function. Paper presented at FQMT'13: Frontiers of Quantum and Mesoscopic Thermodynamics (Prague, Czech Republic, 29 July–3 August 2013). Physica scripta. T, 2015(T 165), Article ID 014043.
Open this publication in new window or tab >>On the representation of the genetic code by the attractors of 2-adic function
2015 (English)In: Physica scripta. T, ISSN 0281-1847, Vol. 2015, no T 165, article id 014043Article in journal (Refereed) Published
Abstract [en]

The genetic code is a map which gives the correspondence between codons in DNA and amino acids. As a continuation of the study made by Khrennikov and Kozyrev on the genetic code, we consider a construction, where amino acids are associated to the attractors of some two-adic function. In this paper, we give an explicit form of representations for the standard nuclear and vertebrate mitochondrial genetics codes. To set these functions we use a van der Put representation. The usage of the van der Put series reduces the complexity of computation for explicit form of the functions for the genetic codes.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2015
Keywords
Genetic code, p-adic, Dynamical systems, van der Put series, Evolution
National Category
Mathematics Genetics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-46679 (URN)10.1088/0031-8949/2015/T165/014043 (DOI)000367396900045 ()2-s2.0-84960411610 (Scopus ID)
Conference
FQMT'13: Frontiers of Quantum and Mesoscopic Thermodynamics (Prague, Czech Republic, 29 July–3 August 2013)
Available from: 2015-10-08 Created: 2015-10-08 Last updated: 2019-02-22Bibliographically approved
Khrennikov, A. & Yurova, E. (2014). Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions. Chaos, Solitons & Fractals, 60, 11-30
Open this publication in new window or tab >>Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions
2014 (English)In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 60, p. 11-30Article in journal (Refereed) Published
Abstract [en]

This paper is devoted to the problem of ergodicity of p-adic dynamical systems. We solved the problem of characterization of ergodicity and measure preserving for (discrete) p-adic dynamical systems for arbitrary prime p for iterations based on 1-Lipschitz functions. This problem was open since long time and only the case p = 2 was investigated in details. We formulated the criteria of ergodicity and measure preserving in terms of coordinate functions corresponding to digits in the canonical expansion of p-adic numbers. (The coordinate representation can be useful, e.g., for applications to cryptography.) Moreover, by using this representation we can consider non-smooth p-adic transformations. The basic technical tools are van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions. As is well known, p-adic spaces have the fractal (although very special) structure. Hence, our study covers a large class of dynamical systems on fractals. Dynamical systems under investigation combine simplicity of the algebraic dynamical structure with very high complexity of behavior.

Keywords
Ergodicity, dynamical systems, van der Put, coordinate functions
National Category
Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-34076 (URN)10.1016/j.chaos.2014.01.001 (DOI)000333719000002 ()2-s2.0-84893440046 (Scopus ID)
Available from: 2014-05-05 Created: 2014-05-05 Last updated: 2017-12-05Bibliographically approved
Anashin, V., Khrennikov, A. & Yurova, E. (2014). Ergodicity criteria for non-expanding transformations of 2-adic spheres. Discrete and Continuous Dynamical Systems, 34(2), 367-377
Open this publication in new window or tab >>Ergodicity criteria for non-expanding transformations of 2-adic spheres
2014 (English)In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 34, no 2, p. 367-377Article in journal (Refereed) Published
Abstract [en]

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems < f; S2-r (a)> on 2-adic spheres S2-r (a) of radius 2(-r), r >= 1, centered at some point a from the ultrametric space of 2-adic integers Z(2). The map f: Z(2) -> Z(2) is assumed to be non-expanding and measure-preserving; that is, f satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and f preserves a natural probability measure on Z(2), the Haar measure mu(2) on Z(2) which is normalized so that mu(2)(Z(2)) = 1.

Keywords
Ergodic theory, 1-Lipschitz dynamics, 2-adic sphere, p-adic analysis
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-30644 (URN)10.3934/dcds.2014.34.367 (DOI)000325646400002 ()2-s2.0-84885971608 (Scopus ID)
Available from: 2013-11-22 Created: 2013-11-22 Last updated: 2017-12-06Bibliographically approved
Yurova Axelsson, E. (2014). On recent results of ergodic property for p-adic dynamical systems. P-Adic Numbers, Ultrametric Analysis, and Applications, 6(3), 235-257
Open this publication in new window or tab >>On recent results of ergodic property for p-adic dynamical systems
2014 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 6, no 3, p. 235-257Article in journal (Refereed) Published
Abstract [en]

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.

Place, publisher, year, edition, pages
Pleiades Publishing, 2014
Keywords
dynamical systems, p-adic, 1-Lipschitz, measure-preserving, ergodicity, spheres, uniformly differentiable
National Category
Algebra and Logic
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-36328 (URN)10.1134/S2070046614030066 (DOI)2-s2.0-84922756866 (Scopus ID)
Available from: 2014-08-11 Created: 2014-08-11 Last updated: 2017-12-05Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-1919-1495

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